HOW MANY SUBSETS WHICH THEIR OPERATIONS WITH A FIXED SUBSET CONTAIN AT LEAST ONE ELEMENT OF A GIVEN COLLECTION?

Huda, Muhammad Nurul
Lestari, Veni Rizki

BAREKENG: Jurnal Ilmu Matematika dan Terapan
Vol 18 No 4 (2024): BAREKENG: Journal of Mathematics and Its Application

We pose the following problem related to binary set operations on finite sets. Given a finite set . Let a binary set operation and be a non-empty collection of non-empty subsets of . For a fixed subset of , where , how many subsets of which their operation with contains at least one element of ?. In this paper, we give the solution of this problem, especially for the subsets of size , using the inclusion-exclusion principle, Corrádi’s lemma, and Bonferroni’s inequality. In this context, the problem is related to determining the degree of nodes in certain graphs, such as graphs constructed with the adjacency rule depends on and the node set is a hypergraph.